PURE MTH 3019 - Complex Analysis III

North Terrace Campus - Semester 1 - 2019

When the real numbers are replaced by the complex numbers in the definition of the derivative of a function, the resulting (complex) differentiable functions turn out to have many remarkable properties not enjoyed by their real analogues. These functions, usually known as holomorphic functions, have numerous applications in areas such as engineering, physics, differential equations and number theory, to name just a few. The focus of this course is on the study of holomorphic functions and their most important basic properties. Topics covered are: Complex numbers and functions; complex limits and differentiability; elementary examples; analytic functions; complex line integrals; Cauchy's theorem and the Cauchy integral formula; Taylor's theorem; zeros of holomorphic functions; Rouche's Theorem; the Open Mapping theorem and Inverse Function theorem; Schwarz' Lemma; automorphisms of the ball, the plane and the Riemann sphere; isolated singularities and their classification; Laurent series; the Residue Theorem; calculation of definite integrals and evaluation of infinite series using residues; Montel's Theorem and the Riemann Mapping Theorem.

  • General Course Information
    Course Details
    Course Code PURE MTH 3019
    Course Complex Analysis III
    Coordinating Unit School of Mathematical Sciences
    Term Semester 1
    Level Undergraduate
    Location/s North Terrace Campus
    Units 3
    Contact Up to 3 hours per week
    Available for Study Abroad and Exchange Y
    Prerequisites MATHS 2100 or MATHS 2101 or MATHS 2202
    Assumed Knowledge MATHS 2101 or MATHS 2202
    Course Description When the real numbers are replaced by the complex numbers in the definition of the derivative of a function, the resulting (complex) differentiable functions turn out to have many remarkable properties not enjoyed by their real analogues. These functions, usually known as holomorphic functions, have numerous applications in areas such as engineering, physics, differential equations and number theory, to name just a few. The focus of this course is on the study of holomorphic functions and their most important basic properties.

    Topics covered are: Complex numbers and functions; complex limits and differentiability; elementary examples; analytic functions; complex line integrals; Cauchy's theorem and the Cauchy integral formula; Taylor's theorem; zeros of holomorphic functions; Rouche's Theorem; the Open Mapping theorem and Inverse Function theorem; Schwarz' Lemma; automorphisms of the ball, the plane and the Riemann sphere; isolated singularities and their classification; Laurent series; the Residue Theorem; calculation of definite integrals and evaluation of infinite series using residues; Montel's Theorem and the Riemann Mapping Theorem.
    Course Staff

    No information currently available.

    Course Timetable

    The full timetable of all activities for this course can be accessed from Course Planner.

  • Learning Outcomes
    Course Learning Outcomes
    1. Demonstrate understanding of the basic concepts underlying complex analyis.

    2. Demonstrate familiarity with a range of examples of these concepts.

    3. Prove basic results in complex analysis.

    4. Apply the methods of complex analysis to evaluate definite integrals and infinite series.

    5. Demonstrate understanding and appreciation of deeper aspects of complex analysis such as the Riemann Mapping theorem.

    6. Demonstrate skills in communicating mathematics orally and in writing.
    University Graduate Attributes

    This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:

    University Graduate Attribute Course Learning Outcome(s)
    Deep discipline knowledge
    • informed and infused by cutting edge research, scaffolded throughout their program of studies
    • acquired from personal interaction with research active educators, from year 1
    • accredited or validated against national or international standards (for relevant programs)
    1,2,3,4,5
    Critical thinking and problem solving
    • steeped in research methods and rigor
    • based on empirical evidence and the scientific approach to knowledge development
    • demonstrated through appropriate and relevant assessment
    1,2,3,4
    Teamwork and communication skills
    • developed from, with, and via the SGDE
    • honed through assessment and practice throughout the program of studies
    • encouraged and valued in all aspects of learning
    6
    Career and leadership readiness
    • technology savvy
    • professional and, where relevant, fully accredited
    • forward thinking and well informed
    • tested and validated by work based experiences
    5
    Self-awareness and emotional intelligence
    • a capacity for self-reflection and a willingness to engage in self-appraisal
    • open to objective and constructive feedback from supervisors and peers
    • able to negotiate difficult social situations, defuse conflict and engage positively in purposeful debate
    6
  • Learning Resources
    Required Resources
    None.
    Recommended Resources
    In increasing order of difficulty, the following books are available in the BSL. The closest to the level of this course is 2.

    1. Churchhill & Brown: Complex Variables and Applications; 517.53 C563
    2. Marsden & Hoffman: Basic Complex Analysis; 517.54 M363b
    3. Conway: Functions of One Complex Variable; 517.53 C767f
    4. Ahlfors: An Introduction to the Theory of Analytic Functions of One Complex Variable; 517.53 A28
    Online Learning
    This course uses MyUni exclusively for providing electronic resources, such as lecture notes, assignment papers, sample solutions, discussion boards, etc. It is recommended that students make appropriate use of these resources.
  • Learning & Teaching Activities
    Learning & Teaching Modes
    Over the course of 30 lectures, the lecturer presents the material to the students and guides them through it. During this time students are expected to engage with the material being presented in lectures, identifying any difficulties that may arise in their understanding of it, and interacting with the lecturer to overcome these difficulties. It is expected that students will attend all lectures, but lectures will be recorded (when facilities allow for this) to help with incidental absences and for revision purposes. In fortnightly tutorials students present their solutions to assigned exercises and discuss them with the lecturer and their peers. Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, allowing them to gauge their progress.
    Workload

    The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

    Activity Quantity Workload Hours
    Lectures 30 90
    Tutorials 5 18
    Assignments 5 50
    Total 158
    Learning Activities Summary
         Lecture Schedule
    Week 1 Complex numbers, functions and differentiation.
    Week 2 Cauchy-Riemann equations. Elementary functions.
    Week 3 Further examples, harmonic functions, complex series.
    Week 4 Analytic functions. Complex antiderivatives.
    Week 5 Integration of complex functions.
    Week 6 Cauchy-Goursat theorem. The Cauchy integral formula.
    Week 7 Consequences of the Cauchy integral formula.
    Week 8 Taylor's theorem. Zeros of holomorphic functions.
    Week 9 The open mapping and inverse function theorems. Isolated singularities of holomorphic functions.
    Week 10 Meromorphic functions, Laurent series; residues.
    Week 11 Applications of residues. Simply connected domains.
    Week 12     The Riemann Mapping theorem.
  • Assessment

    The University's policy on Assessment for Coursework Programs is based on the following four principles:

    1. Assessment must encourage and reinforce learning.
    2. Assessment must enable robust and fair judgements about student performance.
    3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
    4. Assessment must maintain academic standards.

    Assessment Summary
    Assessment Task Task Type Due Weighting Learning Outcomes
    Exam Summative Examination Period 55% All
    Mid-semester test  Summative Week 6 20% 1,2,3,6
    Tutorials Formative and summative Weeks 2,4,8,10,12 5% All
    Assignments Formative and summative Weeks  3,5,7,9,11 20% All
    Assessment Related Requirements
    An aggregate score of 50% is required to pass the course.
    Assessment Detail
    Assessment Set Due Weighting
    Tutorial 1 Week 1 Week 2 1%
    Assignment 1 Week 2 Week 3 4%
    Tutorial 2 Week 3 Week 4 1%
    Assignment 2 Week 4 Week 5 4%
    Midsemester test Week 6 Week 6 20%
    Assignment 3 Week 6 Week 7 4%
    Tutorial 3 Week 7 Week 8 1%
    Assignment 4 Week 8 Week 9 4%
    Tutorial 4 Week 9 Week 10 1%
    Assignment 5 Week 10 Week 11 4%
    Tutorial 5 Week 11 Week 12 1%
    Submission
    Assignments will have a maximum two-week turn-around time for feedback to students.
    Course Grading

    Grades for your performance in this course will be awarded in accordance with the following scheme:

    M10 (Coursework Mark Scheme)
    Grade Mark Description
    FNS   Fail No Submission
    F 1-49 Fail
    P 50-64 Pass
    C 65-74 Credit
    D 75-84 Distinction
    HD 85-100 High Distinction
    CN   Continuing
    NFE   No Formal Examination
    RP   Result Pending

    Further details of the grades/results can be obtained from Examinations.

    Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.

    Final results for this course will be made available through Access Adelaide.

  • Student Feedback

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    SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.

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  • Policies & Guidelines
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